This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the questions: Question 2: Determine the gradient for each of the following equations: a) 2y - 6x = 4 Step 1: Add 6x to both sides of the equation. 2y = 6x + 4 Step 2: Divide both sides by 2. y = (6x)/(2) + (4)/(2) y = 3x + 2 The gradient m is the coefficient of x. The gradient is 3. b) x + 2y = 2 Step 1: Subtract x from both sides of the equation. 2y = -x + 2 Step 2: Divide both sides by 2. y = -(1)/(2)x + (2)/(2) y = -(1)/(2)x + 1 The gradient m is the coefficient of x. The gradient is -(1)/(2). c) 2y = 4x Step 1: Divide both sides by 2. y = (4x)/(2) y = 2x The gradient m is the coefficient of x. The gradient is 2. Question 3: Determine the y-intercept in each of the following equations of straight lines: a) y = 3x - 6 The equation is already in the form y = mx + c, where c is the y-intercept. The y-intercept is -6. b) x + 3y - 4 = 0 Step 1: Rearrange the equation to the form y = mx + c. Subtract x and add 4 to both sides. 3y = -x + 4 Step 2: Divide both sides by 3. y = -(1)/(3)x + (4)/(3) The y-intercept c is the constant term. The y-intercept is (4)/(3). c) (1)/(2)y = x - 3 Step 1: Multiply both sides by 2 to solve for y. y = 2(x - 3) y = 2x - 6 The y-intercept c is the constant term. The y-intercept is -6. d) y = (1)/(2)x + 5 The equation is already in the form y = mx + c, where c is the y-intercept. The y-intercept is 5. Question 4: Determine the gradient of a line with an equation y = 4x + 5. The equation is already in the form y = mx + c, where m is the gradient. The gradient is 4. Question 5: The equation of a line is given as 2x + y = 8. Determine: a) the gradient of the line, Step 1: Rearrange the equation to the form y = mx + c. Subtract 2x from both sides. y = -2x + 8 The gradient m is the coefficient of x. The gradient is -2. b) the value of the y-intercept. From the equation y = -2x + 8, the y-intercept c is the constant term. The y-intercept is 8. Question 6: Given a line with an equation (2)/(3)x + (5)/(7)y = 1. a) Express the equation in the form y = mx + c. Step 1: Subtract (2)/(3)x from both sides of the equation. (5)/(7)y = 1 - (2)/(3)x Step 2: Multiply both sides by (7)/(5) to isolate y. y = (7)/(5) ( 1 - (2)/(3)x ) y = (7)/(5) - (14)/(15)x y = -(14)/(15)x + (7)/(5) The equation in the form y = mx + c is y = -(14)/(15)x + (7)/(5). b) Determine the gradient of the line. From the equation y = -(14)/(15)x + (7)/(5), the gradient m is the coefficient of x. The gradient is -(14)/(15). c) The value of y-intercept. From the equation y = -(14)/(15)x + (7)/(5), the y-intercept c is the constant term. The y-intercept is (7)/(5). What's next?